Symmetric quadrature rules on a triangle

By S. Wandzurat, Hong Xiao. 2003

162 citations recorded by the source corpus.

Abstract We present a class of quadrature rules on triangles in R 2 which, somewhat similar to Gaussian rules on intervals in R 1 , have rapid convergence, positive weights, and symmetry. By a scheme combining simple group theory and numerical optimization, we obtain quadrature rules of this kind up to the order 30 on triangles. This scheme, essentially a formalization and generalization of the approach used by Lyness and Jespersen over 25 years ago, can be easily extended to other regions in R 2 and surfaces in higher dimensions, such as squares, spheres. We present example formulae and relevant numerical results.

Subfields: Applied Mathematics, Computational Mechanics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/6ec907c860ee711ddfcf7520765737eb703e7a45

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In Applied Mathematics

In Computational Mechanics

In Numerical Analysis

Citations in the corpus

High‐order continuous and discontinuous Galerkin methods for wave problems (2013, background): “Here, Wandzurat symmetric rules for triangles are considered, see [59].”
The generation of arbitrary order curved meshes for 3D finite element analysis (2013, methodology): “To produce an accurate and efficient high-order finite element solver, appropriate nodal distributions for the interpolation [3,4,32] and optimal numerical quadratures for the integration [9,35,10] are used.”
A SET OF SYMMETRIC QUADRATURE RULES ON TRIANGLES AND TETRAHEDRA (2009, methodology): “Many methods have been developed for computing quadrature rules; we refer the reader to [1, 2, 3, 4, 5, 6, 7, 8, 9] and the references therein.”
Local maximum‐entropy approximation schemes: a seamless bridge between finite elements and meshfree methods (2006, background): “Symmetric rules ranging from 1 to 175 points per triangle are considered in the calculations [27].”
Generalized Gaussian Quadrature Rules for Discontinuities and Crack Singularities in the Extended Finite Element Method (2010, background): “Wandzura and Xiao [63] and Xiao and Gimbutas [35] applied this algorithm to triangles and squares for polynomial basis functions of very high orders.”
Conforming polygonal finite elements (2004, methodology): “in finite element meshes [11, 12]; suitable in material design [ 13 ] and in the modelling of polycrystalline materials [14]; and will have less sensitivity to lock (unlike lower-order triangular and quadrilateral elements which tend to be stiff) under volume-preserving deformation states which aris”
Generalized Gaussian quadrature rules on arbitrary polygons (2010, background): “Moderate- to high-order efficient quadrature rules have been developed for polynomials on the triangle [4–12].”